Optimal. Leaf size=93 \[ -\frac {2 i b^3 \log (x)}{(a+i)^4}+\frac {2 i b^3 \log (a+b x+i)}{(a+i)^4}+\frac {2 b^2}{(1-i a)^3 x}+\frac {i b}{(a+i)^2 x^2}-\frac {-a+i}{3 (a+i) x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5095, 77} \[ \frac {2 b^2}{(1-i a)^3 x}-\frac {2 i b^3 \log (x)}{(a+i)^4}+\frac {2 i b^3 \log (a+b x+i)}{(a+i)^4}+\frac {i b}{(a+i)^2 x^2}-\frac {-a+i}{3 (a+i) x^3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 5095
Rubi steps
\begin {align*} \int \frac {e^{2 i \tan ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {1+i a+i b x}{x^4 (1-i a-i b x)} \, dx\\ &=\int \left (\frac {i-a}{(i+a) x^4}-\frac {2 i b}{(i+a)^2 x^3}+\frac {2 i b^2}{(i+a)^3 x^2}-\frac {2 i b^3}{(i+a)^4 x}+\frac {2 i b^4}{(i+a)^4 (i+a+b x)}\right ) \, dx\\ &=-\frac {i-a}{3 (i+a) x^3}+\frac {i b}{(i+a)^2 x^2}+\frac {2 b^2}{(1-i a)^3 x}-\frac {2 i b^3 \log (x)}{(i+a)^4}+\frac {2 i b^3 \log (i+a+b x)}{(i+a)^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 88, normalized size = 0.95 \[ \frac {(a+i) \left (a^3+i a^2+3 i a b x+a-6 i b^2 x^2-3 b x+i\right )+6 i b^3 x^3 \log (a+b x+i)-6 i b^3 x^3 \log (x)}{3 (a+i)^4 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 94, normalized size = 1.01 \[ \frac {-6 i \, b^{3} x^{3} \log \relax (x) + 6 i \, b^{3} x^{3} \log \left (\frac {b x + a + i}{b}\right ) - 6 \, {\left (i \, a - 1\right )} b^{2} x^{2} + a^{4} + 2 i \, a^{3} + {\left (3 i \, a^{2} - 6 \, a - 3 i\right )} b x + 2 i \, a - 1}{{\left (3 \, a^{4} + 12 i \, a^{3} - 18 \, a^{2} - 12 i \, a + 3\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 136, normalized size = 1.46 \[ -\frac {2 \, b^{4} \log \left (b x + a + i\right )}{a^{4} b i - 4 \, a^{3} b - 6 \, a^{2} b i + 4 \, a b + b i} + \frac {2 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{4} i - 4 \, a^{3} - 6 \, a^{2} i + 4 \, a + i} + \frac {a^{4} i - 2 \, a^{3} + 6 \, {\left (a b^{2} + b^{2} i\right )} x^{2} - 3 \, {\left (a^{2} b + 2 \, a b i - b\right )} x - 2 \, a - i}{3 \, {\left (a + i\right )}^{4} i x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 560, normalized size = 6.02 \[ \frac {2 b^{3} \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{\left (a^{2}+1\right )^{4}}+\frac {a^{2}}{3 \left (a^{2}+1\right ) x^{3}}+\frac {2 b^{2}}{\left (a^{2}+1\right )^{3} x}-\frac {8 b^{3} \ln \relax (x ) a^{3}}{\left (a^{2}+1\right )^{4}}+\frac {2 b a}{\left (a^{2}+1\right )^{2} x^{2}}-\frac {6 b^{2} a^{2}}{\left (a^{2}+1\right )^{3} x}-\frac {4 b^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a}{\left (a^{2}+1\right )^{4}}+\frac {2 b^{3} \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right ) a^{4}}{\left (a^{2}+1\right )^{4}}-\frac {12 b^{3} \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right ) a^{2}}{\left (a^{2}+1\right )^{4}}+\frac {4 b^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{3}}{\left (a^{2}+1\right )^{4}}+\frac {8 b^{3} \ln \relax (x ) a}{\left (a^{2}+1\right )^{4}}-\frac {1}{3 \left (a^{2}+1\right ) x^{3}}-\frac {2 i b^{3} \ln \relax (x ) a^{4}}{\left (a^{2}+1\right )^{4}}+\frac {12 i b^{3} \ln \relax (x ) a^{2}}{\left (a^{2}+1\right )^{4}}+\frac {i b \,a^{2}}{\left (a^{2}+1\right )^{2} x^{2}}-\frac {2 i b^{2} a^{3}}{\left (a^{2}+1\right )^{3} x}-\frac {6 i b^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{2}}{\left (a^{2}+1\right )^{4}}-\frac {8 i b^{3} \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right ) a^{3}}{\left (a^{2}+1\right )^{4}}+\frac {8 i b^{3} \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right ) a}{\left (a^{2}+1\right )^{4}}+\frac {i b^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{4}}{\left (a^{2}+1\right )^{4}}+\frac {6 i b^{2} a}{\left (a^{2}+1\right )^{3} x}-\frac {2 i b^{3} \ln \relax (x )}{\left (a^{2}+1\right )^{4}}-\frac {2 i a}{3 \left (a^{2}+1\right ) x^{3}}-\frac {i b}{\left (a^{2}+1\right )^{2} x^{2}}+\frac {i b^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{\left (a^{2}+1\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 263, normalized size = 2.83 \[ \frac {{\left (2 \, a^{4} - 8 i \, a^{3} - 12 \, a^{2} + 8 i \, a + 2\right )} b^{3} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1} + \frac {{\left (i \, a^{4} + 4 \, a^{3} - 6 i \, a^{2} - 4 \, a + i\right )} b^{3} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1} + \frac {{\left (-2 i \, a^{4} - 8 \, a^{3} + 12 i \, a^{2} + 8 \, a - 2 i\right )} b^{3} \log \relax (x)}{a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1} + \frac {a^{6} - 2 i \, a^{5} - {\left (6 i \, a^{3} + 18 \, a^{2} - 18 i \, a - 6\right )} b^{2} x^{2} + a^{4} - 4 i \, a^{3} - {\left (-3 i \, a^{4} - 6 \, a^{3} - 6 \, a + 3 i\right )} b x - a^{2} - 2 i \, a - 1}{3 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.72, size = 199, normalized size = 2.14 \[ \frac {\frac {a-\mathrm {i}}{3\,\left (a+1{}\mathrm {i}\right )}-\frac {b^2\,x^2\,2{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^3}+\frac {b\,x\,1{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^2}}{x^3}+\frac {b^3\,\mathrm {atanh}\left (\frac {a^4+a^3\,4{}\mathrm {i}-6\,a^2-a\,4{}\mathrm {i}+1}{{\left (a+1{}\mathrm {i}\right )}^4}-\frac {x\,\left (2\,a^{12}\,b^2+12\,a^{10}\,b^2+30\,a^8\,b^2+40\,a^6\,b^2+30\,a^4\,b^2+12\,a^2\,b^2+2\,b^2\right )}{{\left (a+1{}\mathrm {i}\right )}^4\,\left (-b\,a^9+3{}\mathrm {i}\,b\,a^8+8{}\mathrm {i}\,b\,a^6+6\,b\,a^5+6{}\mathrm {i}\,b\,a^4+8\,b\,a^3+3\,b\,a-b\,1{}\mathrm {i}\right )}\right )\,4{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.15, size = 286, normalized size = 3.08 \[ - \frac {2 i b^{3} \log {\left (- \frac {2 a^{5} b^{3}}{\left (a + i\right )^{4}} - \frac {10 i a^{4} b^{3}}{\left (a + i\right )^{4}} + \frac {20 a^{3} b^{3}}{\left (a + i\right )^{4}} + \frac {20 i a^{2} b^{3}}{\left (a + i\right )^{4}} + 2 a b^{3} - \frac {10 a b^{3}}{\left (a + i\right )^{4}} + 4 b^{4} x + 2 i b^{3} - \frac {2 i b^{3}}{\left (a + i\right )^{4}} \right )}}{\left (a + i\right )^{4}} + \frac {2 i b^{3} \log {\left (\frac {2 a^{5} b^{3}}{\left (a + i\right )^{4}} + \frac {10 i a^{4} b^{3}}{\left (a + i\right )^{4}} - \frac {20 a^{3} b^{3}}{\left (a + i\right )^{4}} - \frac {20 i a^{2} b^{3}}{\left (a + i\right )^{4}} + 2 a b^{3} + \frac {10 a b^{3}}{\left (a + i\right )^{4}} + 4 b^{4} x + 2 i b^{3} + \frac {2 i b^{3}}{\left (a + i\right )^{4}} \right )}}{\left (a + i\right )^{4}} - \frac {- i a^{3} + a^{2} - i a - 6 b^{2} x^{2} + x \left (3 a b + 3 i b\right ) + 1}{x^{3} \left (3 i a^{3} - 9 a^{2} - 9 i a + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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